Problem: A circle has a radius of $2$. An arc in this circle has a central angle of $105^\circ$. What is the length of the arc? ${4\pi}$ ${105^\circ}$ $\color{#DF0030}{\dfrac{7}{6}\pi}$ ${2}$
Answer: First, calculate the circumference of the circle. $c = 2\pi r = 2\pi (2) = 4\pi$ The ratio between the arc's central angle $\theta$ and $360^\circ$ is equal to the ratio between the arc length $s$ and the circle's circumference $c$ $\dfrac{\theta}{360^\circ} = \dfrac{s}{c}$ $\dfrac{105^\circ}{360^\circ} = \dfrac{s}{4\pi}$ $\dfrac{7}{24} = \dfrac{s}{4\pi}$ $\dfrac{7}{24} \times 4\pi = s$ $\dfrac{7}{6}\pi = s$